1. Convolution Based Profile Fitting with TOPAS
Arnt Kern
Title picture can be exchanged in the masterslide.
2011 TOPAS User's Meeting

2. TOPAS Convolution Based Profile Fitting
TOPAS features a direct convolution approach with all parameters refineable
Choice of empirical or physically meaningful profile fitting
Virtually any peak shape, angle dependence as well as hkl dependence (anisotropic line broadening) can normally be described using a minimum number of profile parameters
Quantitative microstructure analysis
"Double-Voigt Approch" (e.g. Balzar, 1999)
"WPPM" (Scardi & Leoni, 2001, 2004; David et al., 2010; Scardi et al. 2010)
Anisotropic strain (Leineweber, 2010, 2011)
...
2011 TOPAS User's Meeting

3. TOPAS Convolution Based Profile Fitting
Kern, A., Cheary, R.W., Coelho, A.A. (2004): Convolution Based Profile Fitting. Diffraction Analysis of the Microstructure of Materials. Editors: Mittemeijer, E.J. & Scardi, P. Springer Series in Materials Science, Vol. 68, 552 pages, ISBN:
Cheary, R.W., Coelho, A.A. & Cline, J.P. (2004): Fundamental Parameters Line Profile Fitting in Laboratory Diffractometers. J. Res. Natl. Inst. Stand. Technol., 109, 1-25.
Kern, A. (2008): Convolution Based Profile Fitting. Principles and Applications of Powder Diffraction. Editors: Clearfield, A., Bhuvanesh N. & Reibenspies J. Blackwell Publishers, 400 pages. ISBN:
2011 TOPAS User's Meeting

4. TOPAS Convolution Based Profile Fitting
Methodology
Physically meaningless parametrization of observed line profile shapes
Explicit discrimination of instrument and sample contributions to observed line profile shapes
Y(2q) = (W  G)  S, with (W  G) = I
(Well known approach latest since Klug & Alexander, 1954)
Measured I: Based on a standard reference material
Calculated I: Fundamental Parameters Approach
Y(2q): Observed line profile shape, I: Instrument function, W: Source emission profile, G: Geometric instrument aberations
2011 TOPAS User's Meeting

5. TOPAS Convolution Based Profile Fitting
Profile shape functions (PSFs) are generated by convoluting functions together to form the observed profile shape:
Y(2q) = F1(2q)  F2(2q)  ...  Fi(2q)  ...  Fn(2q)
In general any combination of appropriate functions for Fi(2q) may be used
The functions Fi(2q) can be interpreted as the aberration functions of the diffractometer: FPA
2011 TOPAS User's Meeting

6. Convolution Based Profile Fitting "Joy of Convolution"
The process of convolution is one in which the product of two functions f(2q) and h(2q) is integrated over all space,
y(2q) = f(2q)  h(2q) = f(2q') h(2q - 2q') d(2q')
where
y(2q) is the convolution product,
2q' is the variable of integration in the same 2q domain, and
 denotes the convolution process.
In simple terms, convolution can be understood as "blending" one function with another, producing a kind of very general "moving average". The convoluted function is obtained by setting down the origin of the first function in every possible position of the second, multiplying the values of both functions in each position, and taking the sum of all operations.
See e.g.
2011 TOPAS User's Meeting

7. Convolution Based Profile Fitting "Joy of Convolution"
Demo
TOPAS
2011 TOPAS User's Meeting

8. TOPAS Convolution Synthesis of Line ProfilesW
Fi(2q)
Y(2q)
Y(2q) = F1(2q)  F2(2q)  ...  Fi(2q)  ...  Fn(2q)
2011 TOPAS User's Meeting

9. TOPAS Convolution Based Profile Fitting
Applicability
Virtually any peak shape, angle dependence as well as hkl dependence (anisotropic line broadening) can normally be described using a minimum number of profile parameters
Powder Diffraction
Laboratory and synchrotron X-ray data
CW and TOF neutron data
Other X-ray applications
PDF, SAXS
Other XY data types
XRF, Raman, NMR, OES, DTA / DSC ... (peak positions, intensities)
2011 TOPAS User's Meeting

10. Source Emission Profiles
2011 TOPAS User's Meeting

11. Source Emission Profiles
Generation of the emission profile is the first step in peak generation. An emission profile comprises EMk lines, each of which is a Voigt function comprising the following parameters:
la: Area under the emission profile line (relative intensities for more than one line; the line with the largest la will be used for d-spacing calculations by default)
lo: Wavelength in [Å] of the emission profile line
lh: Lorentzian HW of the emission profile line in [mili-Å] that is convoluted into the emission profile line
lg: Gaussian HW of the emission profile line in [mili-Å] that is convoluted into the emission profile line
Source emission profiles for most common sealed tube / rotating anode materials are found in the "/lam" directory
Instruments with crystal monochromators may require (re)determination of their source emission profile
Synchrotron and neutron sources: User-defined
2011 TOPAS User's Meeting

12. Source Emission Profiles Cu Ka
Cu Ka spectrum fitted by four Lorentzians (dotted lines)
CuKa4_Holzer.lam
Härtwig et al. (1993)
2011 TOPAS User's Meeting

13. Source Emission Profiles Cu Ka Satellites ("CuKa3")
Cu Ka spectrum is better represented using five Lorentzians
CuKa5_Berger.lam
From Cheary et al. (2004)
2011 TOPAS User's Meeting

14. Source Emission Profiles Cu Ka
Two examples for differently sophisticated Cu Ka emission profiles:
1) By default, d-values are calculated from the line with the largest la
2) Approximates modeling in traditional software
3) Intensity ratio Ka1/Ka2 is 0.514
4) Assumes same shape and width for Ka1 and Ka2  misfit!
Emission profile
Emission lines
lo [Å]
la [%]
lh [mili-Å]
CuKa5_Berger
Satellites
1.59
3.6854
Ka1a )
0.4370
Ka1b
7.62
0.6000
Ka2a
25.17
0.5200
Ka2b
8.71
0.6200
CuKa2_analyt 2)
Ka1
, 3)
0.5 4)
Ka2 )
2011 TOPAS User's Meeting

15. Source Emission Profiles Cu Ka Satellites ("CuKa3")
LAM cursor for CuKa5_Berger.lam
Size-Strain Round Robin, Balzar (2000); "sharp data", Le Bail.
2011 TOPAS User's Meeting

16. Source Emission Profiles Finite X-Ray Source Width (Tube Tails)
Intensity scan with 50 mm wide slit of an image formed through a 10 mm pinhole in platinum of the 0.4 mm wide long fine focus in a Cu anode. X-ray tube set at 40kV, 40 mA.
From Cheary et al. (2004)
2011 TOPAS User's Meeting

17. Source Emission Profiles Finite X-Ray Source Width (Tube Tails)
Cu Ka1,2
Cu Ka3
Tube Tail
Tube Tail
SQRT scale
2011 TOPAS User's Meeting

18. Source Emission Profiles Absorption Edge Modelling
Cu Ka3
Cu Ka1,2
Remnant Cu Kß
Remnant
Bremsstrahlung
Ni Abs. Edge
Tube Tail
Tube Tail
SQRT scale
Absorption Edge Modelling

19. TOPAS Convolution Based Profile Fitting
Methodology
Empirical parametrization of observed line profile shapes Profile parameters have NO physical meaning
Explicit discrimination of instrument and sample contributions
Measured Instrument Function Sample related profile parameters have a physical meaning: Absorption, microstructure (size, strain, ...)...
Calculated Instrument Function All profile parameters have a physical meaning (Fundamental Parameters Approach)
2011 TOPAS User's Meeting

20. 1. Empirical parametrization of observed line profile shapes
2011 TOPAS User's Meeting

21. Convolution Based Profile Fitting 1. Empirical Parametrization
Convolute any appropriate functions to achieve a best fit
Minimize the number of functions / function parameters to minimize parameter correlation
Note:
An excellent approach for all profile fit applications using any instrument, if micro-structure information is NOT of interest
Refined profile parameters have NO physical meaning
2011 TOPAS User's Meeting

22. Convolution Based Profile Fitting 1. Empirical Parametrization
Knowledge of the most common contributions to line profile shapes and their dependence on angle helps, e.g. for laboratory instruments:
Do it yourself: Trial and error
Two back-to-back circle or exponential functions convoluted on top of a Voigt function (e.g. David & Jorgensen, 1993), each of which parametrized with appropriate dependence on angle and maybe hkl, will fit virtually anything
Contribution
Convolution
Angular dependence
Detector (Slit)
Hat
Constant
Crystallite size
Lorentzian
1/cos(Th)
Strain
Gaussian
Tan(Th)
Axial divergence
Circle
-1/Tan(Th)
2011 TOPAS User's Meeting

23. CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004)
Size-Strain Round Robin, Balzar (2004); "sharp data".

24. CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004)
Virtually any peak shape and angle dependence can normally be described using a minimum number of (refineable) profile parameters
Size-Strain Round Robin, Balzar (2004); "sharp data".

25. 2a. Explicit discrimination of instrument and sample contributions Measured Instrument Functions
2011 TOPAS User's Meeting

26. Convolution Based Profile Fitting 2a. Measured Instrument Functions
Measured Instrument Function Approach:
Step 1: Determine an instrument function
Obtain data from a suited standard reference material, e.g.:
Reflection: LaB6 (NIST SRM 660a)
Transmission: Si (NIST SRM 640c)
Convolute any appropriate functions to achieve a best fit (Empirical parameterization!, see above)
The number of functions / function parameters is irrelevant, the more the better!
Fix all refineable profile parameters and save an instrument file
Step 2: Refine the actual data using the instrument function
Load the instrument file
Refine on micro-structure parameters as required
2011 TOPAS User's Meeting

27. Convolution Based Profile Fitting 2a. Measured Instrument Functions
Measured Instrument Function Approach:
Note:
An excellent approach for all profile fit applications using any instrument
Micro-structure information can be derived
Allows to deal with higher degrees of peak overlap (with significant impact specifically on quantitative Rietveld analysis)
Uses a minimal number of profile parameters (specimen contributions such as size/strain/...)
Sample related profile parameters have a physical meaning (Absorption, microstructure (size, strain, ...)...
Measured instrument functions need redetermination after instrument alignment / change of configuration
2011 TOPAS User's Meeting

28. CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004)
Size-Strain Round Robin, Balzar et al. (2004); "broad data".

29. 2b. Explicit discrimination of instrument and sample contributions Calculated Instrument Functions (Fundamental Parameters Approach)
2011 TOPAS User's Meeting

30. Convolution Based Profile Fitting 2b. Calculated Instrument Functions
Calculated Instrument Function Approach:
Physically based diffractometer models are used by convoluting instrument aberation functions.
This can be done for most powder diffractometer configurations including
Conventional divergent beam instruments (most convenient!)
Parallel beam instruments
Instruments used for asymmetric diffraction, (e.g. instruments equipped with position sensitive detectors)
 Fundamental Parameters Approach
2011 TOPAS User's Meeting

31. Convolution Based Profile Fitting 2b. Calculated Instrument Functions
Calculated Instrument Function Approach:
Note:
The ideal approach for all profile fit applications - not only if micro-structure information is of interest
The Bragg Brentano Geometry is the most convenient
Micro-structure information can be derived
Allows to deal with higher degrees of peak overlap (with significant impact specifically on quantitative Rietveld analysis)
Uses a minimal number of profile parameters (normally only specimen contributions such are refined)
All profile parameters have a physical meaning
2011 TOPAS User's Meeting

32. Example: Bragg-Brentano Geometry Beam Path and Optics
2011 TOPAS User's Meeting

33. CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004)
Laboratory X-ray data (D8 ADVANCE):
Instrument function (FPA)
1 Lorentzian function: Crystallite size broadening
Total number of refineable parameters: 1

34. CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004)
D8 ADVANCE, "sharp data"

35. CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004)
2011 TOPAS User's Meeting

36. CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004)
TOPAS results submitted (Kern, A.)